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Fluctuation limit of branching processes with immigration and estimation of the means

Part of: Markov processes Inference from stochastic processes Parametric inference

Published online by Cambridge University Press:  01 July 2016

M. Ispány ,
G. Pap  and
M. C. A. van Zuijlen
M. Ispány*
Affiliation:
University of Debrecen
G. Pap*
Affiliation:
University of Debrecen
M. C. A. van Zuijlen*
Affiliation:
Radboud University Nijmegen
*
∗ Postal address: Faculty of Informatics, University of Debrecen, Pf. 12, H-4010 Debrecen, Hungary.
∗ Postal address: Faculty of Informatics, University of Debrecen, Pf. 12, H-4010 Debrecen, Hungary.
∗∗∗∗ Postal address: Faculty of Mathematics, Radboud University Nijmegen, Toernooiveld 1, 6525 ED Nijmegen, The Netherlands. Email address: [email protected]
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Abstract

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We investigate a sequence of Galton-Watson branching processes with immigration, where the offspring mean tends to its critical value 1 and the offspring variance tends to 0. It is shown that the fluctuation limit is an Ornstein-Uhlenbeck-type process. As a consequence, in contrast to the case in which the offspring variance tends to a positive limit, it transpires that the conditional least-squares estimator of the offspring mean is asymptotically normal. The norming factor is n3/2, in contrast to both the subcritical case, in which it is n1/2, and the nearly critical case with positive limiting offspring variance, in which it is n.

Keywords

Conditional least-squares estimatorfluctuation limitnearly critical Galton-Watson branching process with immigrationOrnstein-Uhlenbeck-type process

MSC classification

Primary: 62F12: Asymptotic properties of estimators
Secondary: 60J80: Branching processes (Galton-Watson, birth-and-death, etc.) 60J60: Diffusion processes 62M05: Markov processes
Type
General Applied Probability
Information
Advances in Applied Probability , Volume 37 , Issue 2 , June 2005 , pp. 523 - 538
Copyright
Copyright © Applied Probability Trust 2005 

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